1. Network Flow
CSE 373
Data Structures

2. Sources and Sinks S T
4/7/2019
CSE 373 AU Network Flow

3. Flow Conservation 14 + 10 = 6 + 11 + 7 6 14 11 10 7 4/7/2019
CSE 373 AU Network Flow

4. Capacities/Saturation
7/9 T
7/10
7/7
7/11
flow / capacity
4/7/2019
CSE 373 AU Network Flow

5. Flow Cancellation 8 5 = 3
4/7/2019
CSE 373 AU Network Flow

6. Max Flow? A 0/12 B 0/20 0/16 0/4 S T 0/10 0/7 0/9 0/13 0/4 C D 0/14
4/7/2019
CSE 373 AU Network Flow

7. A Flow A 12/12 B 19/20 11/16 1/4 S T 0/10 7/7 0/9 12/13 4/4 C D 11/14
4/7/2019
CSE 373 AU Network Flow

8. A Flow saturated 12 + 7 = 19 + 0 A 12/12 B 19/20 11/16 1/4 S T 0/10
7/7
0/9
saturated
12/13
4/4
saturated C D
11/14 =
4/7/2019
CSE 373 AU Network Flow

9. Cuts 23/29 A 12/12 B 19/20 11/16 1/4 S T 0/10 7/7 0/9 12/13 4/4 C D
11/14
Remove edges until S is in one connected
component, and T in the other
23/29
4/7/2019
CSE 373 AU Network Flow

10. Cuts 23/26 A 12/12 B 19/20 11/16 1/4 S T 0/10 7/7 0/9 12/13 4/4 C D
11/14
23/26
Only count forward edges
4/7/2019
CSE 373 AU Network Flow

11. Cuts 23/31 A 12/12 B 19/20 11/16 1/4 S T 0/10 7/7 0/9 12/13 4/4 C D
11/14
23/31
4/7/2019
CSE 373 AU Network Flow

12. Min Cut 23/23 A 12/12 B 19/20 11/16 1/4 S T 0/10 7/7 0/9 12/13 4/4 C D
11/14
23/23
4/7/2019
CSE 373 AU Network Flow

13. Max Flow = Min Cut 23 23 A 12/12 B 19/20 11/16 1/4 S T 0/10 7/7 0/9
12/13
4/4 C D
11/14
No more capacity along cut to take more flow
4/7/2019
CSE 373 AU Network Flow

14. Ford-Fulkerson Method
Single source, single sink
Several algorithms are implementations of this method
Requires integer capacities for steady progress
Fastest implementation computes max flow in O(V^3) time
4/7/2019
CSE 373 AU Network Flow

15. Ford-Fulkerson Method
Initialize flow f to 0
While there exists an augmenting path p
Augment flow f along p
Return f
4/7/2019
CSE 373 AU Network Flow

16. An Augmenting Path How much can be sent along the path? A 0/12 B 0/20
0/16
0/4 S T
0/10
0/7
0/9
0/13
0/4 C D
0/14
How much can be sent along the path?
4/7/2019
CSE 373 AU Network Flow

17. An Augmenting Path Find a path with DFS or BFS. bottleneck A 12/12 B
12/20
12/16
0/4 S T
0/10
0/7
0/9
0/13
0/4 C D
0/14
Find a path with DFS or BFS.
4/7/2019
CSE 373 AU Network Flow

18. Another Augmenting Path
12/12 B
12/20
12/16
0/4 S T
0/10
0/7
0/9
0/13
0/4 C D
0/14
How much can be sent along the path?
4/7/2019
CSE 373 AU Network Flow

19. Another Augmenting Path
12/12 B
bottleneck
12+4=16/20
12+4=16/16
0/4 S T
4/10
4/7
0/9
0/13
0/4 C D
4/14
How much can be sent along the path?
4/7/2019
CSE 373 AU Network Flow

20. Residual Graphs How much more flow can be sent? 4/12 A B 4/16 0/20 0/4
0/10
0/7
4/9
0/13
4/4 C D
4/14
12-4=8 A B 4
16-4=12 20 4 4 S T 10 4 7
9-4=5 13 4 4 C D
4/7/2019
CSE 373 AU Network Flow
14-4=10
How much more flow can be sent?

21. Residual Graphs S A C B D T 4/16 0/13 0/10 0/4 4/9 4/14 4/12 0/20 0/7
4/4 8 A B 4 12 20 4 4 S T 10 4 7 5 13 4 S A C B D T
4/16
4/13
0/10
0/4
0/9
4/14
4/12
4/20
0/7
4/4 4 C D 10 8 A B 4 12 16 4 4 4 S T 10 7 9 4 9 4
Can cancel out flow (BC)
4/7/2019
CSE 373 AU Network Flow 4 C D 10

22. Residual Graphs S A C B D T 4/16 0/13 0/10 0/4 4/9 4/14 4/12 0/20 0/7
4/4 8 A B 4 12 12 4 4 4 S T 10 7 9 4 9 4
12/12 4 A B C D
12/16
12/20 10
0/4
0/10 S T
0/7
0/9 A B
4/13 12
4/4 C D 4 8
4/14 12 12 4 S T 10 7 9 4 9 4
4/7/2019
CSE 373 AU Network Flow 4 C D 10

23. Residual Graphs S A C B D T 4/16 0/13 0/10 0/4 4/9 4/14 4/12 0/20 0/7
4/4 A B 12 4 8 12 12 4 S T 10 7 9 4 9 4
12/12 4 A B C D
12/16 10
19/20
0/4
0/10 S T
7/7
0/9 A B
11/13 12
4/4 C D 4 1
11/14 19 12 4 S T 10 7 9 11 2 4
4/7/2019
CSE 373 AU Network Flow 11 C D 3

24. Residual Graphs 12/12 A B 12/16 19/20 0/4 0/10 S T 7/7 0/9 11/13 4/4 C
11/14 A B 12 4 1 19 12 4 S T 10 7 9 11 2 4 11 C D 3
No more augmenting paths: Max Flow!
Reachable vertices from source form one set, other vertices the other set.
Edges in the middle are the cut.
4/7/2019
CSE 373 AU Network Flow

25. Linear Programming: Max Flow
Variables:
SA, SB, AB, AT, BT
(amount of flow through the edge)
Constraints:
capacity
0 <= SA, SB, AB, AT, BT
SA <= 16
SB <= 13
AB <= 10
AT <= 4
BT <= 20
flow conservation
AS = AB + AT
AB + SB = BT
Objective:
max ( SA + SB + AB + AT + BT ) A
0/4
0/16 S T
0/10
0/13 B
0/20
4/7/2019
CSE 373 AU Network Flow

26. Linear Programming: Min Cut
Variables:
SA, SB, AB, AT, BT (not cut = 0, cut = 1)
AinS, BinS (not in S = 0, in S = 1)
Constraints:
0 <= SA, SB, AB, AT, BT
1 <= SA + AinS (A in S or SA is cut)
1 <= SB + BinS (B in S or SB is cut)
0 <= AB - AinS + BinS
(A in S but not B means AB is cut)
0 <= AT - AinS (A in S means AT is cut)
0 <= BT - BinS (B in S means BT is cut)
Objective:
min (16 SA + 4 AT + 10 AB + 13 SB + 20 BT) A
0/4
0/16 S T
0/10
0/13 B
0/20
4/7/2019
CSE 373 AU Network Flow